Effects of M-Truncated Derivative and Multiplicative Noise on the Exact Solutions of the Breaking Soliton Equation

Abstract

In this article, the fractional–space stochastic (2+1)-dimensional breaking soliton equation (SFSBSE) is taken into account in the sense of M-Truncated derivative. To get the exact solutions to the SFSBSE, we use the modified F-expansion method. There are several varieties of obtained exact solutions, including trigonometric and hyperbolic functions. The attained solutions of the SFSBSE established in this paper extend a number of previously attained results. Moreover, in order to clarify the influence of multiplicative noise and M-Truncated derivative on the behavior and symmetry of the solutions for the SFSBSE, we employ Matlab to plot three-dimensional and two-dimensional diagrams of the exact fractional–stochastic solutions achieved here. In general, a noise term that destroy the symmetry of the solutions increases the solution’s stability.

1. Introduction

Fractional differential equations (FDEs) are applied to analyze a diverse range of physical phenomena in mathematical biology, signal processing, engineering disciplines, electromagnetic theory, and other scientific studies [1,2,3,4,5]. Furthermore, the fractional-order derivative may be used to depict a broad range of scientific phenomena, such as heat, elasticity, fluid dynamics, quantum mechanics, gravity, electrodynamics, sound electrostatics, and so on. Because of the significance of the fractional-order derivative, numerous definitions have been proposed, including the Riemann-Liouville, Caputo, He’s fractional derivative, beta derivative, truncated M-fractional derivative, the Weyl derivative, Riesz, Grunwald-Letnikov, and conformable fractional definitions [6,7,8,9,10,11,12,13].

Sousa et al. [13] have proposed a new fractional derivative titled truncated M-fractional derivative. From here, we define the truncated M-fractional derivative for function $u:[0,\infty )\to \mathbb{R}$ of order $\alpha \in (0,1)$ as follows:

$${\mathbb{T}}_{M,z}^{i,\alpha ,\beta }u\left(z\right)=\underset{h\to 0}{lim}\frac{u\left(z{+}_i{E}_{\beta }\left(h{z}^{-\alpha }\right)\right)-u\left(z\right)}{h},$$

where the truncated Mittag-Leffler function ${}_i{E}_{\beta }\left(z\right),$ for $\beta >0$ and $z\in \mathbb{C}$, is defined as

$${}_i{E}_{\beta }\left(z\right)=\sum _{k=0}^i\frac{z^k}{\mathsf{\Gamma }(\beta k+1)}.$$

For any constants a and b, the truncated M-fractional derivative fulfills the following features:

$$\begin{array}{ccc}\hfill \left(1\right){\mathbb{T}}_{M,z}^{i,\alpha ,\beta }(au+bv)& =& a{\mathbb{T}}_{M,z}^{i,\alpha ,\beta }\left(u\right)+b{\mathbb{T}}_{M,z}^{i,\alpha ,\beta }\left(v\right),\left(2\right){\mathbb{T}}_{M,z}^{i,\alpha ,\beta }\left(z^{\nu }\right)=\frac{\nu }{\mathsf{\Gamma }(\beta +1)}{z}^{\nu -\alpha },\hfill \\ \hfill \left(3\right){\mathbb{T}}_{M,z}^{i,\alpha ,\beta }\left(uv\right)& =& u{\mathbb{T}}_{M,z}^{i,\alpha ,\beta }v+v{\mathbb{T}}_{M,z}^{i,\alpha ,\beta }u,\left(4\right){\mathbb{T}}_{M,z}^{i,\alpha ,\beta }\left(u\right)\left(z\right)=\frac{z^{1-\alpha }}{\mathsf{\Gamma }(\beta +1)}\frac{du}{dz},\hfill \\ \hfill \left(5\right){\mathbb{T}}_{M,z}^{i,\alpha ,\beta }(u\circ v)\left(z\right)& =& u^{{}^{\prime }}\left(v\left(z\right)\right){\mathbb{T}}_{M,z}^{i,\alpha ,\beta }v\left(z\right).\hfill \end{array}$$

Due to the significance of FDEs, a number of strong and effective techniques have been created to determine the exact solutions to these equations. Several of these techniques are $(G^{\prime }/G)$-expansion [14,15], Jacobi elliptic function [16], perturbation [17,18], $exp(-\phi (\varsigma \left)\right)$-expansion [19], sine-cosine [20,21] Hirota’s [22], tanh-sech [23,24], Lie symmetry method [25,26,27], Riccati-Bernoulli sub-ODE [28], etc.

On the other hand, in biology, physics, climatic dynamics, geophysics, and other domains, the significance of adding random effects in the study, prediction, and modeling of complex events has acquired considerable interest. In the presence of noise or fluctuations, partial differential equations are perfect mathematical concepts for expressing complex systems [29,30].

Recent study on stochastic FDEs has been examined, for example, [31,32,33,34,35,36], and the references therein. Consequently, it appears that investigating models of FDEs with some random force is more essential. Therefore, we look at the SFSBSE in this type [37]:

$${\mathbb{T}}_{M,x}^{0,\alpha ,\beta }{\psi }_t-4{\mathbb{T}}_{M,x}^{0,2\alpha ,\beta }\left({\mathbb{T}}_{M,y}^{0,\alpha ,\beta }\psi \right)-2{\mathbb{T}}_{M,x}^{0,2\alpha ,\beta }\psi {\mathbb{T}}_{M,y}^{0,\alpha ,\beta }\psi +{\mathbb{T}}_{M,x}^{0,3\alpha ,\beta }\left({\mathbb{T}}_{M,y}^{0,\alpha ,\beta }\psi \right)=\sigma {\mathbb{T}}_{M,x}^{0,\alpha ,\beta }\psi {\mathcal{W}}_t,$$

(1)

where $\psi$ is a real stochastic function of the variable $x,y$and$t$, $\sigma$ is the noise strength, and $\mathcal{W}$ is a one dimensional Wiener process.

The breaking soliton equations (BSE) explain the (2+1)-dimensional interaction of a Riemann wave moving down the y-axis and a long wave moving along the x-axis. The BSE (1) with $\alpha =1,\beta =0$ and $\sigma =0$ is utilized to clarify the plasma physics, fluid wave leading flow, and hydrodynamic problem of shallow-water waves. Many researchers have derived exact solution to the deterministic breaking soliton problem using different approaches including tanh–coth [38], Hirota bilinear [39], the improved $\left(\frac{G^{\prime }}{G}\right)$-expansion and the extended tanh-method [40], the three-wave methods [41], the Jacobi elliptic functions [42], $\left(\frac{G^{\prime }}{G}\right)$-expansion [43], generalized auxiliary equation [44], and the Riccati equation [45]. The exact solutions to the fractional-stochastic breaking soliton equations with the truncated M-fractional derivative and stochastic term have still not been established.

This work aims to elucidate the analytical solutions of the SFSBSE (1). We use the modified F-expansion method to get various kinds of solutions, including the hyperbolic, rational and trigonometric functions. The acquired exact solutions of the SFSBSE (1) established in this paper extend a number of previously attained results, such as the result described in [38,40]. Additionally, the MATLAB tool was used to show certain graphical representations, and we analyzed the influence of the truncated M-fractional derivative and stochastic term on the analytical solutions of the SFSBSE (1).

The structure of this article is as follows: To obtain the SFSBSE’s (1) wave equation, go to the next section. In Section 3, we find the exact solutions of the SFSBSE (1). In Section 4, we investigate how the truncated M-fractional derivative and stochastic term affect the solutions of the SFSBSE (1). Finally, the conclusions of this paper are presented.

2. The SFSBSE’s Wave Equation

The wave equation of the SFSBSE is attained utilizing the next wave transformation

$$\psi (x,y,t)=u\left(\zeta \right)e^{(\sigma \mathcal{W}\left(t\right)-\frac{{\sigma }^2}{2}t)},\zeta =\frac{\mathsf{\Gamma }(\beta +1)}{\alpha }[{\zeta }_1{x}^{\alpha }+{\zeta }_2{y}^{\alpha }]+{\zeta }_{3}t,$$

(2)

where the real function u is a deterministic, ${\left({\zeta }_i\right)}_{i=1,2,3}$ are constants. We note that

$$\begin{array}{ccc}\hfill {\psi }_t& =& ({\zeta }_3{u}^{\prime }+\sigma u{\mathcal{W}}_t+\frac{1}{2}{\sigma }^{2}u-\frac{1}{2}{\sigma }^{2}u)e^{(\sigma \mathcal{W}\left(t\right)-\frac{{\sigma }^2}{2}t)},\hfill \\ \hfill {\mathbb{T}}_{M,x}^{0,\alpha ,\beta }{\psi }_t& =& ({\zeta }_3{\zeta }_1{u}^{″}+\sigma {\zeta }_1{u}^{\prime }{\mathcal{W}}_t)e^{(\sigma \mathcal{W}\left(t\right)-\frac{{\sigma }^2}{2}t)},\hfill \\ \hfill {\mathbb{T}}_{M,y}^{0,\alpha ,\beta }\psi & =& {\zeta }_2{u}^{\prime }{e}^{(\sigma \mathcal{W}\left(t\right)-\frac{{\sigma }^2}{2}t)},{\mathbb{T}}_{M,x}^{0,\alpha ,\beta }{\mathbb{T}}_{M,y}^{0,\alpha ,\beta }\psi ={\zeta }_1{\zeta }_2{u}^{″}{e}^{(\sigma \mathcal{W}\left(t\right)-\frac{{\sigma }^2}{2}t)},\hfill \\ \hfill {\mathbb{T}}_{M,x}^{0,2\alpha ,\beta }\psi & =& {\zeta }_1^2{u}^{″}{e}^{(\sigma \mathcal{W}\left(t\right)-\frac{{\sigma }^2}{2}t)},{\mathbb{T}}_{M,x}^{0,3\alpha ,\beta }{\mathbb{T}}_{M,y}^{0,\alpha ,\beta }\psi ={\zeta }_1^3{\zeta }_2{u}^{⁗}{e}^{(\sigma \mathcal{W}\left(t\right)-\frac{{\sigma }^2}{2}t)}.\hfill \end{array}$$

(3)

Plugging Equation (2) into (1) and utilizing (3), we get

$${\zeta }_3{u}^{″}-6{\zeta }_1{\zeta }_2{u}^{\prime }{u}^{″}{e}^{(\sigma \mathcal{W}\left(t\right)-\frac{{\sigma }^2}{2}t)}+{\zeta }_1^2{\zeta }_2{u}^{⁗}=0.$$

(4)

We obtain by considering expectation on both sides of (4)

$${\zeta }_3{u}^{″}-6{\zeta }_1{\zeta }_2{u}^{\prime }{u}^{″}{e}^{-\frac{{\sigma }^2}{2}t}\mathbb{E}\left(e^{\sigma \mathcal{W}\left(t\right)}\right)+{\zeta }_1^2{\zeta }_2{u}^{⁗}=0,$$

(5)

where the function u is a deterministic. For each real integer $\gamma$ and any Gaussian process Z, we see that

$$\mathbb{E}\left(e^{\gamma Z}\right)=e^{\frac{{\gamma }^2}{2}t}.$$

(6)

Hence, Equation (5) becomes

$${\zeta }_3{u}^{″}-6{\zeta }_1{\zeta }_2{u}^{\prime }{u}^{″}+{\zeta }_1^2{\zeta }_2{u}^{⁗}=0.$$

(7)

Integrating Equation (7) once with regard to $\zeta$ and ignoring the constant of integral, we attain

$${\zeta }_1^2{\zeta }_2{u}^{‴}+{\zeta }_3{u}^{\prime }-3{\zeta }_1{\zeta }_2{\left[u^{\prime }\right]}^2=0.$$

(8)

Setting

$$u^{\prime }=\phi ,$$

(9)

in Equation (8), we have

$${\phi }^{″}+{\hslash }_1\phi +{\hslash }_2{\phi }^2=0,$$

(10)

where

$${\hslash }_1=\frac{{\zeta }_3}{{\zeta }_1^2{\zeta }_2}\mathrm{and}{\hslash }_2=\frac{-3}{{\zeta }_1}.$$

Modified F-Expansion Method

Here, we use the modified F-expansion method. We suppose the solution $\phi$ of Equation (10) takes the form (with $M=2$):

$$\phi \left(\zeta \right)={\ell }_0+{\ell }_{1}F+{\ell }_2{F}^2+\frac{k_1}{F}+\frac{k_1}{F^2},$$

(11)

where F solves

$$F^{\prime }=F^2+\omega ,$$

(12)

where $\omega$ is a real constant. The solutions of Equation (12) are:

$$F\left(\zeta \right)=\sqrt{\omega }tan\left(\sqrt{\omega }\zeta \right)\mathrm{or}F\left(\zeta \right)=-\sqrt{\omega }cot\left(\sqrt{\omega }\zeta \right),$$

(13)

if $\omega >0,$ or

$$F\left(\zeta \right)=-\sqrt{-\omega }tanh\left(\sqrt{-\omega }\zeta \right)\mathrm{or}F\left(\zeta \right)=-\sqrt{-\omega }coth\left(\sqrt{-\omega }\zeta \right),$$

(14)

if $\omega <0,$ or

$$F\left(\zeta \right)=\frac{-1}{\zeta },$$

(15)

if $\omega =0.$

Substituting Equation (11) into Equation (10), we get

$$\begin{array}{ccc}& & (6{\ell }_2+{\hslash }_2{\ell }_2^2)F^4+(2{\ell }_1+2{\hslash }_2{\ell }_1{\ell }_2)F^3+(8\omega {\ell }_2+2{\ell }_0{\ell }_2{\hslash }_2+{\ell }_1^2{\hslash }_2+{\hslash }_1{\ell }_2)F^2\hfill \\ & & (2\omega {\ell }_1+{\hslash }_1{\ell }_1+2{\hslash }_2{\ell }_0{\ell }_1+2{\ell }_2{k}_1)F+(2{\omega }^2{\ell }_2+2k_2+{\hslash }_1{\ell }_0+{\hslash }_2{\ell }_0^2+2{\hslash }_2{\ell }_1{k}_1\hfill \\ & & +2{\hslash }_2{\ell }_2{k}_2)+(2\omega k_1+2{\hslash }_2{\ell }_0{k}_1+2{\hslash }_2{\ell }_1{k}_2+{\hslash }_1{k}_1)F^{-1}+(8\omega k_2+2{\ell }_0{k}_2{\hslash }_2\hfill \\ & & +k_1^2{\hslash }_2+{\hslash }_1{k}_2)F^{-2}+(2k_1{\omega }^2+2{\hslash }_2{k}_1{k}_2)F^{-3}+(6{\omega }^2{k}_2+{\hslash }_2{k}_2^2)F^{-4}=0\hfill \end{array}$$

Putting the coefficients of each power of F to zero:

$$6{\ell }_2+{\hslash }_2{\ell }_2^2=0,$$

$$2{\ell }_1+2{\hslash }_2{\ell }_1{\ell }_2=0,$$

$$8\omega {\ell }_2+2{\ell }_0{\ell }_2{\hslash }_2+{\ell }_1^2{\hslash }_2+{\hslash }_1{\ell }_2=0,$$

$$2\omega {\ell }_1+{\hslash }_1{\ell }_1+2{\hslash }_2{\ell }_0{\ell }_1+2{\ell }_2{k}_1=0,$$

$$2{\omega }^2{\ell }_2+2k_2+{\hslash }_1{\ell }_0+{\hslash }_2{\ell }_0^2+2{\hslash }_2{\ell }_1{k}_1+2{\hslash }_2{\ell }_2{k}_2=0,$$

$$2\omega k_1+2{\hslash }_2{\ell }_0{k}_1+2{\hslash }_2{\ell }_1{k}_2+{\hslash }_1{k}_1=0,$$

$$8\omega k_2+2{\ell }_0{k}_2{\hslash }_2+k_1^2{\hslash }_2+{\hslash }_1{k}_2=0,$$

$$2k_1{\omega }^2+2{\hslash }_2{k}_1{k}_2=0,$$

and

$$6{\omega }^2{k}_2+{\hslash }_2{k}_2^2=0.$$

We attain a four different sets by solving these equations as follows:

First set:

$${\ell }_0=\frac{-6\omega }{{\hslash }_2},{\ell }_1=0,{\ell }_2=\frac{-6}{{\hslash }_2},k_1=0,k_2=0.$$

(16)

Second set:

$${\ell }_0=\frac{-2\omega }{{\hslash }_2},{\ell }_1=0,{\ell }_2=\frac{-6}{{\hslash }_2},k_1=0,k_2=0.$$

(17)

Third set:

$${\ell }_0=\frac{-12\omega }{{\hslash }_2},{\ell }_1=0,{\ell }_2=\frac{-6}{{\hslash }_2},k_1=0,k_2=\frac{-6{\omega }^2}{{\hslash }_2}.$$

(18)

Fourth set:

$${\ell }_0=\frac{8\omega }{{\hslash }_2},{\ell }_1=0,{\ell }_2=\frac{-6}{{\hslash }_2},k_1=0,k_2=\frac{-6{\omega }^2}{{\hslash }_2}.$$

(19)

First set: The solution of Equation (10) is

$$\phi \left(\zeta \right)=\frac{-6\omega }{{\hslash }_2}-\frac{6}{{\hslash }_2}{F}^2\left(\zeta \right).$$

For $F\left(\zeta \right)$, there are three cases:

Case 1: If $\omega >0,$ then by using (13) we get

$$\phi \left(\zeta \right)=\frac{-6\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{tan}^2\left(\sqrt{\omega }\zeta \right)=-\frac{6\omega }{{\hslash }_2}{sec}^2\left(\sqrt{\omega }\zeta \right),$$

and

$$\phi \left(\zeta \right)=\frac{-6\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{cot}^2\left(\sqrt{\omega }\zeta \right)=\frac{-6\omega }{{\hslash }_2}{csc}^2\left(\sqrt{\omega }\zeta \right).$$

Thus, the FSQZKE (1), by using Equations (2) and (9), has the solutions

$$\psi (x,y,t)=-\frac{6\sqrt{\omega }}{{\hslash }_2}tan\left(\sqrt{\omega }\zeta \right)e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)},$$

(20)

and

$$\psi (x,y,t)=\frac{6\sqrt{\omega }}{{\hslash }_2}cot\left(\sqrt{\omega }\zeta \right)e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)},$$

(21)

where $\zeta =\frac{\mathsf{\Gamma }(\beta +1)}{\alpha }({\zeta }_1{x}^{\alpha }+{\zeta }_2{y}^{\alpha })+{\zeta }_{3}t.$

Case 2: If $\omega <0,$ then by using (14) we get

$$\phi \left(\zeta \right)=\frac{-6\omega }{{\hslash }_2}+\frac{6\omega }{{\hslash }_2}{tanh}^2\left(\sqrt{-\omega }\zeta \right)=\frac{-6\omega }{{\hslash }_2}{\sec \mathrm{h}}^2\left(\sqrt{-\omega }\zeta \right),$$

and

$$\phi \left(\zeta \right)=\frac{-6\omega }{{\hslash }_2}+\frac{6\omega }{{\hslash }_2}{coth}^2\left(\sqrt{-\omega }\zeta \right)=\frac{6\omega }{{\hslash }_2}{\csc \mathrm{h}}^2\left(\sqrt{-\omega }\zeta \right).$$

Thus, the FSQZKE (1), by using Equations (2) and (9), has the solutions

$$\psi (x,y,t)=\frac{-6\sqrt{-\omega }}{{\hslash }_2}tanh\left(\sqrt{-\omega }\zeta \right)e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)},$$

(22)

and

$$\psi (x,y,t)=\frac{-6\sqrt{-\omega }}{{\hslash }_2}coth\left(\sqrt{-\omega }\zeta \right)e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)},$$

(23)

where $\zeta =\frac{\mathsf{\Gamma }(\beta +1)}{\alpha }({\zeta }_1{x}^{\alpha }+{\zeta }_2{y}^{\alpha })+{\zeta }_{3}t.$

Case 3: If $\omega =0,$ then by using (15) we get

$$\phi \left(\zeta \right)=\frac{6}{{\hslash }_2}\frac{1}{{\zeta }^2}.$$

Thus, the FSQZKE (1), by using Equations (2) and (9), has the solution

$$\psi (x,y,t)=[-\frac{6}{{\hslash }_2}\frac{1}{\zeta }]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)},$$

(24)

where $\zeta =\frac{\mathsf{\Gamma }(\beta +1)}{\alpha }({\zeta }_1{x}^{\alpha }+{\zeta }_2{y}^{\alpha })+{\zeta }_{3}t.$

Second set: The solution of Equation (10) in this case is

$$\phi \left(\zeta \right)=\frac{-2\omega }{{\hslash }_2}-\frac{6}{{\hslash }_2}{F}^2\left(\zeta \right)$$

For $F\left(\zeta \right)$, there are three cases:

Case 1: If $\omega >0,$ then by using (13) we get

$$\phi \left(\zeta \right)=\frac{-2\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{tan}^2\left(\sqrt{\omega }\zeta \right)=\frac{4\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{sec}^2\left(\sqrt{\omega }\zeta \right),$$

and

$$\phi \left(\zeta \right)=\frac{-2\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{cot}^2\left(\sqrt{\omega }\zeta \right)=\frac{4\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{csc}^2\left(\sqrt{\omega }\zeta \right).$$

Thus, the FSQZKE (1), by using Equations (2) and (9), has the solutions

$$\psi (x,y,t)=[\frac{4\omega }{{\hslash }_2}\zeta -\frac{6\sqrt{\omega }}{{\hslash }_2}tan\left(\sqrt{\omega }\zeta \right)]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)},$$

(25)

and

$$\psi (x,y,t)=[\frac{4\omega }{{\hslash }_2}\zeta +\frac{6\sqrt{\omega }}{{\hslash }_2}cot\left(\sqrt{\omega }\zeta \right)]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)},$$

(26)

where $\zeta =\frac{\mathsf{\Gamma }(\beta +1)}{\alpha }({\zeta }_1{x}^{\alpha }+{\zeta }_2{y}^{\alpha })+{\zeta }_{3}t.$

Case 2: If $\omega <0,$ then by using (14) we get

$$\phi \left(\zeta \right)=\frac{-2\omega }{{\hslash }_2}+\frac{6\omega }{{\hslash }_2}{tanh}^2\left(\sqrt{-\omega }\zeta \right)=\frac{4\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{\sec \mathrm{h}}^2\left(\sqrt{-\omega }\zeta \right),$$

and

$$\phi \left(\zeta \right)=\frac{-2\omega }{{\hslash }_2}+\frac{6\omega }{{\hslash }_2}{coth}^2\left(\sqrt{-\omega }\zeta \right)=\frac{4\omega }{{\hslash }_2}+\frac{6\omega }{{\hslash }_2}{\csc \mathrm{h}}^2\left(\sqrt{-\omega }\zeta \right).$$

Thus, the FSQZKE (1), by using Equations (2) and (9), has the solutions

$$\psi (x,y,t)=[\frac{4\omega }{{\hslash }_2}\zeta +\frac{6\sqrt{-\omega }}{{\hslash }_2}tanh\left(\sqrt{-\omega }\zeta \right)]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)},$$

(27)

and

$$\psi (x,y,t)=[\frac{4\omega }{{\hslash }_2}\zeta -\frac{6\sqrt{-\omega }}{{\hslash }_2}coth\left(\sqrt{-\omega }\zeta \right)]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)}.$$

(28)

Case 3: If $\omega =0,$ then by using (15) we get

$$\phi \left(\zeta \right)=\frac{6}{{\hslash }_2}\frac{1}{{\zeta }^2}.$$

Thus, the solution of FSQZKE (1) is

$$\psi (x,y,t)=\frac{-6}{{\hslash }_2}\frac{1}{\zeta }{e}^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)}.$$

(29)

Third set: The solution of Equation (10) is

$$\phi \left(\zeta \right)=\frac{-12\omega }{{\hslash }_2}-\frac{6}{{\hslash }_2}{F}^2\left(\zeta \right)-\frac{6{\omega }^2}{{\hslash }_2}{F}^{-2}\left(\zeta \right).$$

For $F\left(\zeta \right)$, there are three cases:

Case 1: If $\omega >0,$ then by using (13) we get

$$\begin{array}{ccc}\hfill \phi \left(\zeta \right)& =& \frac{-12\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{tan}^2\left(\sqrt{\omega }\zeta \right)-\frac{6\omega }{{\hslash }_2}{cot}^2\left(\sqrt{\omega }\zeta \right)\hfill \\ & =& -\frac{6\omega }{{\hslash }_2}[{sec}^2\left(\sqrt{\omega }\zeta \right)+{csc}^2\left(\sqrt{\omega }\zeta \right)].\hfill \end{array}$$

Thus, the FSQZKE (1), by using Equations (2) and (9), has the solution

$$\psi (x,y,t)=-\frac{6\sqrt{\omega }}{{\hslash }_2}[tan\left(\sqrt{\omega }\zeta \right)-cot\left(\sqrt{\omega }\zeta \right)]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)}.$$

(30)

Case 2: If $\omega <0,$ then by using (14) we get

$$\begin{array}{ccc}\hfill \phi \left(\zeta \right)& =& \frac{-12\omega }{{\hslash }_2}+\frac{6\omega }{{\hslash }_2}{tanh}^2\left(\sqrt{-\omega }\zeta \right)+\frac{6\omega }{{\hslash }_2}{coth}^2\left(\sqrt{-\omega }\zeta \right)\hfill \\ & =& \frac{-6\omega }{{\hslash }_2}[{\sec \mathrm{h}}^2\left(\sqrt{-\omega }\zeta \right)-{\csc \mathrm{h}}^2\left(\sqrt{-\omega }\zeta \right)].\hfill \end{array}$$

Thus, the solution of FSQZKE (1), by using Equations (2) and (9), is

$$\psi (x,y,t)=\frac{-6\sqrt{-\omega }}{{\hslash }_2}[tanh\left(\sqrt{-\omega }\zeta \right)+coth\left(\sqrt{-\omega }\zeta \right)]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)}.$$

(31)

Case 3: If $\omega =0,$ then by using (15) we get

$$\phi \left(\zeta \right)=\frac{6}{{\hslash }_2}\frac{1}{{\zeta }^2}+\frac{6}{{\hslash }_2}{\zeta }^2.$$

Thus, the solution of FSQZKE (1), by using Equations (2) and (9), is

$$\psi (x,y,t)=\frac{6}{{\hslash }_2}[\frac{-1}{\zeta }+\frac{1}{3}{\zeta }^3]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)}.$$

(32)

Fourth set: The solution of Equation (10) is

$$\phi \left(\zeta \right)=\frac{8\omega }{{\hslash }_2}-\frac{6}{{\hslash }_2}{F}^2\left(\zeta \right)-\frac{6{\omega }^2}{{\hslash }_2}{F}^{-2}\left(\zeta \right).$$

For $F\left(\zeta \right)$, there are three cases:

Case 1: If $\omega >0,$ then by using (13) we get

$$\begin{array}{ccc}\hfill \phi \left(\zeta \right)& =& \frac{8\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{tan}^2\left(\sqrt{\omega }\zeta \right)-\frac{6\omega }{{\hslash }_2}{cot}^2\left(\sqrt{\omega }\zeta \right)\hfill \\ & =& \frac{20\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{sec}^2\left(\sqrt{-\omega }\zeta \right)-\frac{6\omega }{{\hslash }_2}{csc}^2\left(\sqrt{-\omega }\zeta \right).\hfill \end{array}$$

Thus, the FSQZKE (1), by using Equations (2) and (9), has the solutions

$$\psi (x,y,t)=[\frac{20\omega }{{\hslash }_2}\zeta -\frac{6\sqrt{\omega }}{{\hslash }_2}tan\left(\sqrt{\omega }\zeta \right)+\frac{6\sqrt{\omega }}{{\hslash }_2}cot\left(\sqrt{\omega }\zeta \right)]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)}.$$

(33)

Case 2: If $\omega <0,$ then by using (14) we get

$$\begin{array}{ccc}\hfill \phi \left(\zeta \right)& =& \frac{8\omega }{{\hslash }_2}+\frac{6\omega }{{\hslash }_2}{tanh}^2\left(\sqrt{-\omega }\zeta \right)+\frac{6\omega }{{\hslash }_2}{coth}^2\left(\sqrt{-\omega }\zeta \right)\hfill \\ & =& \frac{20\omega }{{\hslash }_2}-\frac{6\omega }{{\hslash }_2}{\sec \mathrm{h}}^2\left(\sqrt{-\omega }\zeta \right)+\frac{6\omega }{{\hslash }_2}{\csc \mathrm{h}}^2\left(\sqrt{-\omega }\zeta \right).\hfill \end{array}$$

Thus, the solution of FSQZKE (1), by using Equations (2) and (9), is

$$\psi (x,y,t)=[\frac{20\omega }{{\hslash }_2}\zeta -\frac{6\sqrt{-\omega }}{{\hslash }_2}tanh\left(\sqrt{-\omega }\zeta \right)-\frac{6\sqrt{-\omega }}{{\hslash }_2}coth\left(\sqrt{-\omega }\zeta \right)]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)}.$$

(34)

Case 3: If $\omega =0,$ then by using (15) we get

$$\phi \left(\zeta \right)=\frac{6}{{\hslash }_2}\frac{1}{{\zeta }^2}+\frac{6}{{\hslash }_2}{\zeta }^2.$$

Thus, the FSQZKE (1), by using Equations (2) and (9), has the solution

$$\psi (x,y,t)=\frac{6}{{\hslash }_2}[\frac{-1}{\zeta }+\frac{1}{3}{\zeta }^3]e^{(\sigma \mathcal{W}\left(t\right)-\frac{1}{2}{\sigma }^{2}t)},$$

(35)

where $\zeta =\frac{\mathsf{\Gamma }(\beta +1)}{\alpha }({\zeta }_1{x}^{\alpha }+{\zeta }_2{y}^{\alpha })+{\zeta }_{3}t.$

Remark 1. 

If we put $\sigma =\beta =0,$ and $\alpha =1$ in Equations (20)–(23), we have the similar results indicated in [40].

Remark 2. 

If we put $\sigma =\beta =0,$ and $\alpha =1$ in Equations (22), (23) and (31), we obtain similar solutions, which are indicated in [38].

3. The Impact of M-Truncated Derivative and Noise

Here, we study the impact of the M-truncated derivative and multiplicative noise on the solutions of the SFSBSE (1). We exhibit various figures to show how the behavior of these solutions is affected by the M-truncated derivative and noise. The solutions of (1) are plotted via the MATLAB software. The fixed parameters ${\zeta }_1={\zeta }_2=1$ and ${\zeta }_3=-2$, $y=0.5$, as well as $0\le x\le 5$ and $0\le t\le 3,$, are as follows:

First the impact of noise:

When the noise is ignored (i.e., at $\sigma =0$), we can conclude from Figure 1 and Figure 2 that there are various forms of solutions, including periodic solutions, dark solutions, and others. When noise is considered, the surface after minor transit patterns flattens out and gains strength by $\sigma =1,2$. This shows that the stochastic term has an effect on the SFSBSE solutions and stabilizes the solutions around zero.

Second the impact of the M-truncated derivative:

Finally, we deduced from Figure 3 and Figure 4 that the curves do not overlap. Moreover, the solutions shift to the right as the order of the fractional derivative decreases.

4. Conclusions

In this article, we obtained the exact solutions of the fractional–space stochastic (2+1)-dimensional breaking soliton equation (SFSBSE) in the sense of being M-truncated (1). We applied the modified F-expansion method to obtain the exact solutions of the SFSBSE. The modified F-expansion method works consistently and successfully, and it may be used to solve a wide range of partial differential equations. This method gives us a variety of solutions, including rational, trigonometric, and hyperbolic functions, but it excludes elliptic ones. The obtained exact solutions are far more accurate and necessary in comprehending certain exceedingly complicated physical events. We expanded certain previously obtained results, such as those published in [38,40]. Finally, we demonstrated how the solutions of SFSBSE were impacted by the M-truncated derivative and noise.